Finite Group Databasebeta

← #13267064

Subgroups — #13267064

Order 1792.

310 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3249C211yesmaximal, characteristic
2#11728832C212yesmaximal
3#11728216C211yesmaximal, characteristic
4#11725582C212yesmaximal
5#11725515C211yesmaximal, characteristic
6C7×#533—21no—
7C7×#477—22no—
8C7×#467—21no—
9C7×#467C2212yes—
10C7×#462—22no—
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×#6392C2211yescharacteristic
13#89700—21no—
14#89359—22no—
15#89346—21no—
16#89346C2211yescharacteristic
17C22⋊(C7⋊C16)C2212yes—
18#89285—22no—
19#12289—71nomaximal, Sylow
20C2×C7×Q16—21no—
21C2×C7×Q16D412yes—
22C22×C7×C8D411yescharacteristic
23C22×C7×C8C2311yescharacteristic
24C7×C4⋊Q8—21no—
25C7×C4⋊Q8—21no—
26C7×#125—41no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—41no—
29C4×C7×Q8—22no—
30C4×C7×Q8—41no—
31C7×#117—41no—
32C2×C42×C7D411yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×Dic7⋊C4—41no—
36C7×#107—21no—
37C7×#107D412yes—
38C7×C4⋊C8—22no—
39C7×#103—22no—
40C7×#103—22no—
41C7⋊#95—41no—
42#6392—22no—
43C2×C7⋊C16—22no—
44#3249D711yescharacteristic
45#2235—72no—
46#2086—71no—
47#1348—72no—
48#1322—71no—
49C2×C7×Q8—22no—
50C2×C7×Q8—41no—
51C22×C4×C7D811yescharacteristic
52C22×C4×C7D811yescharacteristic
53C22×C4×C7C2×D411yescharacteristic
54C22×Dic7—41no—
55C22×Dic7—41no—
56C7×Q16—42no—
57C7×Q16—81no—
58C2×C7×C8—21no—
59C2×C7×C8—41no—
60C2×C7×C8—41no—
61C2×C7×C8C2×D412yes—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—42no—
66C7×C4⋊C4—41no—
67C7×C4⋊C4—42no—
68C7×C22⋊C4—42no—
69C42×C7—21no—
70C42×C7—21no—
71C42×C7—42no—
72C4⋊Dic7—22no—
73C4⋊Dic7—41no—
74C4⋊Dic7—41no—
75C4⋊Dic7—41no—
76Dic7⋊C4—82no—
77C7⋊C16—42no—
78#533—21no—
79#477—22no—
80#467—21no—
81#467C2×D712yes—
82#462—22no—
83C2×#107—71no—
84C2×C4⋊C8C2×D711yescharacteristic
85#389—141no—
86#367—142no—
87#358—142no—
88C22⋊C16—72no—
89#340—71no—
90#340—141no—
91C23×C7C2×D811yescharacteristic
92C7×Q8—42no—
93C7×Q8—42no—
94C7×Q8—82no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—41no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7SD3212yes—
104C2×C4×C7#12011yescharacteristic
105C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
106C2×Dic7—42no—
107C2×Dic7—81no—
108C2×Dic7—81no—
109C2×Dic7—81no—
110C2×Dic7—81no—
111C2×Dic7—81no—
112C7×C8—22no—
113C7×C8—41no—
114C2×Q16—21no—
115C2×Q16C22⋊D712yes—
116C22×C8C22⋊D711yescharacteristic
117C22×C8C22×D711yescharacteristic
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C4×Q8—22no—
124C4×Q8—41no—
125#117—41no—
126C2×C4⋊C4—141no—
127C2×C4⋊C4—141no—
128C2×C4⋊C4—281no—
129C2×C42C22⋊D711yescharacteristic
130C2×C16—142no—
131#107—21no—
132#107—142no—
133#107C22⋊D712yes—
134C4⋊C8—22no—
135#103—22no—
136#103—22no—
137#95—281no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7#44711yescharacteristic
141C22×C7C2×SD3211yescharacteristic
142C22×C7#51011yescharacteristic
143C4×C7—21no—
144C4×C7—22no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7—82no—
148C4×C7#45611yescharacteristic
149C4×C7#45811yescharacteristic
150Dic7—82no—
151Dic7—161no—
152C2×Q8—22no—
153C2×Q8—41no—
154C22×C4—281no—
155C22×C4—281no—
156C22×C4#114111yescharacteristic
157C22×C4#114111yescharacteristic
158C22×C4C2×C22⋊D711yescharacteristic
159Q16—42no—
160Q16—81no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8C2×C22⋊D712yes—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—142no—
172C4⋊C4—281no—
173C4⋊C4—281no—
174C4⋊C4—281no—
175C4⋊C4—562no—
176C22⋊C4—42no—
177C42—21no—
178C42—21no—
179C42—42no—
180C16—282no—
181C2×C7—21no—
182C2×C7—21no—
183C2×C7#161011yescharacteristic
184C2×C7#218511yescharacteristic
185C2×C7#219111yescharacteristic
186C23C2×#114111yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—282no—
199C2×C4—561no—
200C2×C4—561no—
201C2×C4—561no—
202C2×C4—561no—
203C2×C4—561no—
204C2×C4#640212yes—
205C2×C4#649511yescharacteristic
206C2×C4#651611yescharacteristic
207C8—22no—
208C8—41no—
209C7#1228911yescharacteristic, complemented, Sylow
210C22—21no—
211C22—21no—
212C22C2×#640211yescharacteristic
213C22#8995111yescharacteristic
214C22#8998911yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—562no—
221C4—1121no—
222C4#9001511yescharacteristic
223C4#9001811yescharacteristic
224C2—21no—
225C2—21no—
226C2#1172962811yescharacteristic
227C2#1172981711yescharacteristic
228C2#1172986711yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#3249C2
socleC22×C7#447
prime radicalC23×C7C2×D8
hypercenter#3249D7
hypocenterC7#12289
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
209. C7#12289semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#32492
#117288322
#117282162
#117255822
#117255152
#122897